This concerns the coadjoint representation of certain semi-direct products M×GL(n) (here M is a matrix space where GL(n) acts) and more particularly that of the affine group. In the later case, we have obtained an explicit inverse of an orbital map (corresponding to a point whose isotropy subgroup is trivial) and this explicit inverse is used to solve certain questions of the invariant theory concerning the affine group and certain of its sub-groups. For example, we get the rational GL(n)-invariants of a number a covariant or contravariant vectors and a number of matrices (in which GL(n) acts by the adjoint representation.